MATH 361 Exam 1, Fall 2001

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1. Solve the equation: ty′ + 2y = sin t, y(π/2) = 1

Ans: .

2. Solve the equation: y′ = 2x/(y + x^{2}y), y(0) = −2

Ans: .

3. Solve the equation:

(cos x + ln y + 4)dx + (\frac{x}{y} + e^{y}) dy = 0

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4. Sketch the solutions of y′ = y^{2} − 9y, satisfying each of the initial conditions: a) y(0) = 2 , y(0) = 5 and y(0) = −3. Classify the equilibrium solutions as asympotically stable, unstable or semistable.

5. A tank originally contains 100 gal of fresh water. Water containing 1/2 lb of salt per gallon is poured into the tank at a rate of 2 gal/min and the mixture is allowed to leave at the same rate. Find the amount of salt in the tank after 15 min.